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The main characteristic of the controlled PN approach to supervisory control is the fact that no transition structure for a controller is given. The control law is a function of the actual marking of the net, but need to be computed at each step, whereas if a transition structure is given for a supervisor, a closed-loop model can be constructed and analyzed to ensure that it has the desired properties.

2.3 Petri Net Languages

Baker [Baker 72] is one of the first to introduce the basic idea of associating a language with a Petri net and using this language to describe the behavior of the net. In his Master’s thesis [Baker 73], Petri nets are considered in the context of formal language theory, and their prefix language is studied.

In Hack [Hack 75a, Hack 75b], two basic language classes are distinguished: the prefix language and the marked language. Different languages which result from different types of transition labeling are also considered. For each of these classes, simple closure properties, characterizations and decidability problems (e.g., for membership, emptiness, and finiteness) are obtained.

A survey and tutorial on Petri net languages is presented in [Peterson 81] Chapter 6. Here the definition and classification of these languages is given; the closure properties of Petri net languages under several form of composition (union, intersection, concatenation, concurrency, and substitution) and under some operations (reversal, complement, and indefinite concatenation) are examined; the relationship between Petri net languages and other classes of formal languages are investigated.

Another comprehensive tutorial on Petri net languages is a paper by Jantzen [Jantzen 87]. Closure properties and relationship among the different classes of Petri net languages are reported with additional results, not presented by Peterson.

Parigot and Peltz [Parigot 86] have characterized PN languages in terms of logical formalism. The language power of Petri nets is shown to be greater than that of finite automata as a result of the additional ability of Petri nets to test if a string of parentheses is well formed. The logical formalism may be used to prove that a given language is a Petri net language and to construct a Petri net having a given finite behavior. Peltz [Peltz 86] has also extended the study to the infinite sequential behavior of Petri nets.

2.4 Incidence Matrix Analysis of Petri Nets

Incidence matrix and related analysis techniques are used by several authors to validate properties of P/T nets.

Colom [Colom 89] develops a methodology for the verification of assertions on P/T nets in terms of markings and firing count vectors. This approach is extremely general, i.e., can be applied to any P/T net but, unfortunately, can only guarantee necessary or sufficient conditions. There exists assertions, such as determining if a marking is a home state, for which neither a necessary nor a sufficient condition can be given. Within this framework, Silva and Colom [Silva 89a] give algorithms to efficiently compute the structural synchronic invariants of P/T nets.

Berthomieu [Berthomieu 87] uses the set of linear equations given by the P-semiflows to represent the space of reachable markings. In this case, only properties that depend on the markings can be proved. However, the author also shows that it is possible to prove some properties that depend on the firing count vector by adding to the net new places whose marking indicates the number of times a given transition has fired.

Johnen [Johnen 87] uses an hybrid approach, based partly on incidence matrix analysis and partly on the analysis of the state space, to verify that a given marking is a home state.

of the subnets. The authors introduce a modified Petri net model [Beck 86] more suitable for the modeling of manufacturing systems by means of the synthesis approach they propose.

Datta and Gosh [Datta 84, Datta 86] present a technique for the modular synthesis of PN. The nets being considered are regular nets, i.e., nets where forks and joins are symmetrical, and where the subnets between a fork and a join are marked graphs. The modules are connected through a set of input/output places, that act as mailboxes. Finally, it is possible to give rules to connect the modules in a way that preserves liveness and boundedness.

Zhou [Zhou 90] has studied a formal methodology for the synthesis of Petri nets in manufacturing. The approach followed is called hybrid, in the sense that it is partially top-down and partially bottom-up. The author has also discussed how liveness properties for systems with shared resources depend on the value of the initial marking of the net.

Koh [Koh 92] has studied the compositions of live and bounded circuits along paths. The composition allows the fusion of overlapping paths, and has been extended to colored Petri nets. The author has studied under which conditions the composed system is live and bounded.

The reversal of N = (P; T; P re; P ost) is the net NR = (P; T; P ost; P re), i.e., a new net where the direction of all arcs of N is reversed.

A state machine is a P/T net such that each transition has exactly one input arc and one output arc. A marked graph is a P/T net such that each place has exactly one input arc and one output arc.

A P/T net is connected if in the underlying graph1 there exists a path (not necessarily directed) from any vertex to all others; strongly connected if there exists a directed path from any vertex to all others; acyclic if no directed path forms a cycle.

3.1.2 Marked Nets

A marking is a vector M : P ! IN that assigns to each place of a P/T net a non-negative integer number of tokens, represented by black dots. M(p) denotes the number of tokens assigned by marking M to place p. The set of all markings defined on a net N = (P; T; P re; P ost) is INjP j.

A net system or marked net hN; M0i is a net N with an initial marking M0.

A transition t 2 T is enabled at a marking M if M ¸ P re(¢; t). If t is enabled at M, then t may fire yielding a new marking M0 with M0 = M + C(¢; t). We will write M [ti M0 to denote that t may fire at M yielding M0.

A firing sequence from M0 is a (possibly empty) sequence of transitions ¾ = t1 : : : tk such that M0 [t1i M1 [t2i M2 ¢ ¢ ¢ [tki Mk. We will also write M0 [¾i Mk to denote that we may fire ¾ at M0 yielding Mk.

A marking M is reachable in hN; M0i if there exists a firing sequence ¾ such that M0 [¾i M. Given a marked net hN; M0i, the set of firing sequences (also called language of the net) is denoted L(N; M0) and the set of reachable markings (also called reachability set of the net) is

denoted R(N; M0).

Let marking M be reachable from marking M0 by firing a sequence of transitions ¾. Then the following state equation is satisfied: M = M0 + C ¢ ~¾, where ~¾ : T ! IN is a vector of non-negative integers, called the firing count vector . ~¾(t) represents the number of times transition t appears in ¾. The set of markings M such that there exists a vector ~¾ satisfying the previous state equation is called the potentially reachable set and is denoted P R(N; M0). Note that P R(N; M0) ¶ R(N; M0).

The firing subnet given by a firing count vector ~¾ consists of all the transitions t 3 ~¾(t) ¸ 0, and of their input and output places.

Let B be a basis of P-semiflows of the net N. A marking M 2 P R(N; M0) satisfies the following system of equations: BT ¢M = BT ¢M0. The set of markings satisfying the previous system of equations is denoted P RB(N; M0) [Colom 89]. Note that P RB(N; M0) ¶ P R(N; M0).

Amarked net hN; M0i is:

²Bounded, if there exists a nonnegative integer k such that M(p) · k for every place p and for every marking M 2 R(N; M0);

²Safe (or 1-bounded), if M(p) · 1 for every place p and for every marking M 2 R(N; M0);

²Conservative, if there exists a vector of positive integers Y such that Y T ¢ M = Y T ¢ M0 for every marking M 2 R(N; M0);

²Live, if from every marking M 2 R(N; M0) there exists a firing sequence containing all transitions;

1The graph underlying a Petri net has as vertices places and transitions, and as directed edges the arcs specified by

P re and P ost.

Given a DES G = (N; `; M0; F ), the L-type language of G is

Lm(G) = f`(¾) 2 Σ¤ j ¾ 2 T ¤; M0 [¾i M 2 F g; and the P-type language of G is

L(G) = f`(¾) 2 Σ¤ j ¾ 2 T ¤; 9M0 3 M0 [¾i M0g:

Note that in this definition of labeled net we are assuming that ` is a ¸-free labeling function, according to the terminology of [Peterson 81], i.e., no transition is labeled with the empty string ¸ and two (or more) transitions may have the same label. The classes of L-type and P-type languages generated by labeled nets is denoted L and P respectively.

A deterministic PN generator [Jantzen 87] is such that the string of events generated from

istic. The classes of L-type and P-type PN languages generated by deterministic PN generators are denoted, respectively, Ld and Pd. Note that Pd ½ P and Ld ½ L (strict inclusion) [Jantzen 87].

3.4 Supervisory Control

This section presents the language notation used for Supervisory Control. A more detailed review of Supervisory Control is presented in Appendix B.

Let L be a language on alphabet Σ. Its prefix closure is the set of all prefixes of strings in L: L = f¾ 2 Σ¤ j 9¿ 2 Σ¤ 3 ¾¿ 2 Lg. A language L is said to be closed if L = L.

Two languages L1 and L2 are said to be non-conflicting [Wonham 88b] if L1 \ L2 = L1 \L2. In the following, let G = (N; `; M0; F ) be a DES with alphabet of events Σ.

G is nonblocking if any string that belongs to its closed behavior may be completed to a string that belongs to the marked behavior. A deterministic DES is non-blocking if and only if

L(G) = Lm(G).

A language K ½ Σ¤ is said to be Lm(G)-closed if K\Lm(G) = K\Lm(G). In the case that K µ Lm(G), this definition is reduced to the usual definition of Lm(G)-closed: K = K\Lm(G) [Ramadge 89b].

The alphabet of events Σ is partitioned into two disjoint subsets: Σc, the set of controllable events, and Σu, the set of uncontrollable events. The controllable events may be disabled by a controlling agent in order to restrict the behavior of the system within a legal behavior, while uncontrollable events may never be disabled.

A language K ½ Σ¤ is said to be controllable with respect to L(G) if KΣu \ L(G) µ K [Ramadge 87]. The set of all languages controllable wrt L(G) is denoted C(G).

If a language L ½ Σ¤ is not controllable with respect to L(G) we may compute its supremal controllable sublanguage [Wonham 87] defined as: L" = supfK µ L j K 2 C(G)g.

Figure 4.1: Blocking system in Example 4.1.

Figure 4.2: Trimmed system in Example 4.1.

marked behavior Lm(G) and closed behavior L(G) ¾ Lm(G) we want to modify the structure of the net so that no blocking markings may be reached, i.e., so that L(G) = Lm(G).

On a simple model such as a state machine this may be done, trivially, by removing all states that are reachable but not coreachable (i.e., no final state may be reached from them) and all their input and output transitions.

On Petri net models the trimming may be more complex. In Chapter 5 is discussed the case of conservative nets. If the Petri net is conservative the trimming may be done without major changes of the net structure, in the sense that we have to add new arcs and possibly duplicate transitions without introducing new places.

As the following example shows, unbounded Petri net models may require more extensive changes of the net structure.

Example 4.1. Let G be the PN generator in Figure 4.1, with M0 = (100)T and set of final markings F = f(001)T g. The behaviors of this system are: Lm(G) = famcbn j m = rn; n ¸ 0g

and L(G) = famcbn j m ¸ rn; n ¸ 0g. The system is clearly blocking, since the string w = ar+1c 2 L(G), for example, cannot be completed to a string in Lm(G) (assuming r > 1).

To avoid reaching a blocking state we require that p2 contain a multiple of r tokens when the transition labeled c is allowed to fire. A possible solution is shown in Figure 4.2, where we have introduced r ¡ 1 new places and new transitions labeled a.

There are some cases, furthermore, in which the trimming of a net system is not possible. The reason for this is given by the following theorem.

Theorem 4.1. There exist L-type Petri net languages whose prefix closure is not a P-type Petri net language, i.e., 9L 2 L 3 L 62 P.

The proof of this theorem will be given by means of the next example and the following proposition.

Figure 4.3: Blocking system G in Example 4.2.

Example 4.2. Let G be the PN generator in Figure 4.3.(a), with M0 = (1000)T and set of final markings F = f(0001)T g. The behaviors of this system are: Lm(G) = fambamb j m ¸ 0g and L(G) = fambanb j m ¸ n ¸ 0g. From the reachability tree of the system, shown in Figure 4.3.(b), it is clear that the system is blocking. To avoid reaching a blocking state we require that p2 be empty before firing the transition inputting into p4. However, since p2 is unbounded this may not be done with a simple P/T structure. It is possible to introduce an inhibitor arc from p2 to the transition inputting into p4. Inhibitor arcs increase the modeling power, and the analysis complexity, of Petri nets to that of a Turing machine, thus we cannot properly consider these models as P/T nets. Petri nets with inhibitory arcs have been studied in the context of Supervisory Control by Sreenivas and Krogh [Sreenivas 92].

We may formally prove that the prefix closure of the marked language of the system G discussed in Example 4.2 is not a P-type Petri net language. The proof is based on the pumping lemma for P-type PN languages, given in [Jantzen 87].

Lemma 4.1 (Pumping lemma). Let L 2 P. Then there exist numbers k; l such that any string w 2 L, with j w j¸ k, has a decomposition w = xyz with 1 ·j y j· l such that xyiz 2 L; 8i ¸ 1.

Proposition 4.1. L = fambamb j m ¸ 0g is not a P-type Petri net language.

We conclude this section with the definition of a new class of L-type Petri net languages whose prefix closure can be generated by a deterministic nonblocking Petri net generator. This class will play an important role in characterizing the existence of Petri net supervisors, as we will discuss in the next section.

Definition 4.1. A language L 2 L (not necessarily deterministic) is said to be deterministic P- closed (DP-closed for short) if and only if its prefix closure is a deterministic P-type Petri net language, i.e., L 2 Pd. The class of DP-closed Petri net languages is denoted LDP .